Invariants
Base field: | $\F_{199}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 51 x + 1041 x^{2} - 10149 x^{3} + 39601 x^{4}$ |
Frobenius angles: | $\pm0.0122497791918$, $\pm0.200340655229$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.10933.1 |
Galois group: | $D_{4}$ |
Jacobians: | $5$ |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $30443$ | $1547813449$ | $62073703293329$ | $2459352248424450925$ | $97393683789182037485648$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $149$ | $39083$ | $7876775$ | $1568225211$ | $312079621604$ | $62103837947231$ | $12358664131961681$ | $2459374187569385731$ | $489415464041408369975$ | $97393677358492410640478$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=88x^6+95x^5+8x^4+34x^3+32x^2+10x+179$
- $y^2=71x^6+172x^5+55x^4+36x^3+30x^2+99x+166$
- $y^2=90x^6+148x^5+142x^4+106x^3+137x^2+194x+88$
- $y^2=61x^6+126x^5+28x^4+150x^3+95x^2+8x+93$
- $y^2=158x^6+196x^5+154x^4+59x^3+63x^2+17x+110$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{199}$.
Endomorphism algebra over $\F_{199}$The endomorphism algebra of this simple isogeny class is 4.0.10933.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.199.bz_bob | $2$ | (not in LMFDB) |